On some questions of topology for -valued fractional Sobolev spaces

$Page 1: On some questions of topology for -valued fractional Sobolev spaces$
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Rev. R. Acad. Cien. Serie A. Mat.VOL. 95 (1), 2001, pp. 121–143Matematica Aplicada / Applied Mathematics

On some questions of topology for �� -valued fractionalSobolev spaces

H. Brezis and P. Mironescu

Abstract. The purpose of this paper is to describe the homotopy classes (i.e., path-connected compo-nents) of the space �� . Here, �� "!#�� , � is a smooth, bounded, connectedopen set in $�% and � �� "&('*),+-� � � ��/.0�� 1 )213&4� a.e. 576Our main results assert that � �� is path-connected if �8!"�:9 while it has the same homotopyclasses as ;=<>�*?�@�� if �8!BA�9 . We also present some results and open problems about density of smoothmaps in �C�� D�E?��F� � .

Sobre algunas cuestiones topologicas para espacios de Sobolevfracionarios con valores en GIH

Resumen. El proposito de este artıculo es describir las clases de homotopıa (i.e., componentes conexaspor arcos) del espacio �� . Aquı, �J�K�=�K��L�=�M!N�O� , � es un abierto, regular, acotado yconexo de $�% y � �� "&('*),+-� � � ��/.0�� 1 )213&4� a.e. 576Nuestros resultados principales establecen que � � � �� es conexo por arcos si �P!-�O9 aunque tengalas mismas clases de homotopıa que ;=<Q�E?�@�� si �P!#AR9 . Tambien presentamos algunos resultados yproblemas abiertos sobre la densidad de aplicaciones regulares de � � � �E?�� .

1. IntroductionThe purpose of this paper is to describe the homotopy classes (i.e., path-connected components) of the spaceSUT V W@XZYJ[ G\H^] . Here, _-`�a�`�bdc � `"ef`dbgc Y is a smooth, bounded, connected open set in hIi andS T V W XjYJ[ G H ]lkUm3nfo S TLV W XZYJ[p ] [@q n q k �srut vwtyxwtOur main results are

Theorem 1 If aef`�z , thenS{T V W|XjYJ[ GIHE] is path-connected.

Theorem 2 If aef}�z , thenS{T V W|XjYJ[ GIHE] and ~�� X>�YJ[ G\HE] have the same homotopy classes in the sense of

[7]. More precisely:a) each nfo S{T V W@XZYJ[ G\H^] is

SUT V W-homotopic to some �No#~�� X �Y�[ GIH*] ;

b) two maps nFc��o�~�� X �YJ[ G\HE] are ~J� -homotopic if and only if they areS{T V W

-homotopic.

Presentado por Jesus Ildefonso DıazRecibido: 8 de Septiembre 2001. Aceptado: 10 de Octubre 2001.Palabras clave / Keywords: homotopy classes, fractional Sobolev spaces, � � -valued maps.Mathematics Subject Classifications: 46E35, 46T10, 46T30, 58D15.c�

2001 Real Academia de Ciencias, Espana.

121


Here a simple consequence of the above results

Corollary 1 If _-`�a�`�bdc � `"ef`db andY

is simply connected, thenS�T V W=XjYJ[ G\HE] is path-connected.

Indeed, when a�eK`�z this is the content of Theorem 1. When a�e�}Uz , we use a) of Theorem 2 to connectn H c�n��o SUT V W@XZYJ[ G\H^] to � H c��o"~�� X>�YJ[ G\HE] ; sinceY

is simply connected, we may write �7��k v>��D� for� ��of~�� X>�YJ[ hl] and then we connect � H to �� via v>�^� � HL�� 8� � �w�/ .When ¡ is a compact connected manifold, the study of the topology of

S H V W@XZYJ[ ¡{] was initiatedin Brezis - Li [7] (see also White [26] for some related questions). In particular, these authors provedTheorems 1 and 2 in the special case a¢k � . The analysis of homotopy classes for an arbitrary manifold ¡and aNk � was subsequently tackled by Hang - Lin [15]. The passage to

S£T V Wintroduces two additional

difficulties:a) when a is not an integer, the

S{T V Wnorm is not “local”;

b) when a�}�z (or more generally a�¤ �I¥ HW ), gluing two maps inS{T V W

does not yield a map inS�T V W

.In our proofs, we exploit in an essential way the fact that the target manifold is G@H . (The case of a

general target is widely open.) In particular, we use the existence of a lifting ofS T V W

unimodular mapswhen a�} � and ae¦}gz (see Bourgain - Brezis - Mironescu [4]). Another important tool is the following

Composition Theorem (Brezis - Mironescu [10]) If §ö©~�� X h [ hª] has bounded derivatives anda�} � , then �O«¬� §-® � is continuous fromS{TLV W=¯�S H V TZW into

SUTLV W.

Remark 1 A very elegant and straightforward proof of this Composition Theorem has been given by V.Maz’ya and T. Shaposhnikova [18].

A related question is the description, when ae�}£z , of the homotopy classes ofS£T V W=XjYJ[ GIHE] in terms of

lifting. Here is a partial result

Theorem 3 We havea) if a�} � c�°±}�² , and z�³Caef`�° , then´ n�µ T V W kUm^n v �¶� [ � o S T V W XjYJ[ h·] ¯¦S H V TjW XjYJ[ h\] x [b) if aef}C° , then ´ nuµ TLV W k�mQn v �� [ � o S T V W XZYJ[ h·] xwt

Theorem 3 is due to Rubinstein - Sternberg [21] in the special case where aNk � c7e�k¸z andY

is thesolid torus in h�¹ .

When _�`da�` � c�°º}d² and z,³dae"`d° , there is no such simple description of´ nuµ T V W . For instance,

using the “non-lifting” results in Bourgain - Brezis - Mironescu [4], it is easy to see that´ � µ T V WB» ¼½ m v �¶� [ � o S TLV W XjYJ[ h·] xDtHere is an example: if °¾kR²�c Y kd¿ H c�_-`ga�` � c � Òef`�bdcuz�³Caef`�² , thena) n XÁÀ ]lk v H�ÂQÃ ÄwÃ ÅNo ´ � µ T V W ;b) there is no � o SUT V W=X ¿ H [ h·] such that nMk v>�� for Æ satisfying ¹*� TjWW ³ÇÆ�` ¹*� TjWTZW .

However, we conjecture the following result

Conjecture 1 Assume that _-`Ca�` � c � Òe�`gbdc�°±}Ç² and zB³�aef`Ç° . Then´ nuµ T V W kdn m v �� [ � o S T V W XjYJ[ h·] xuÈ�É/Ê Ë�tWe will prove below (see Corollary 2) that “half” of Conjecture 1 holds, namely´ nuµ T V WB» n m v �� [ � o S T V W XjYJ[ h·] xuÈ�É/Ê Ë�t

In a different but related direction, we establish some partial results concerning the density of ~-� X �YJ[ G\H^]into

S T V W XZYJ[ G H ] .122

On some questions of topology for � � -valued fractional Sobolev spaces

Theorem 4 We have, for _-`Ca�`gbdc � Òef`�b :a) if aef` � , then ~�� X �YJ[ GIHE] is dense in

S{T V W|XjYJ[ GIHE] ;b) if � ³�ae�`gz�c°Ì}gz , then ~�� X �YJ[ G\HE] is not dense in

S{T V W@XZYJ[ G\HQ] ;c) if ae�}C° , then ~�� X �YJ[ GIHE] is dense in

S{T V W@XZYJ[ G\H^] ;d) if a�} � and ae�}�z , then ~�� X �YJ[ G\HE] is dense in

SUTLV W@XjYJ[ G\HE] .There is only one missing case for which we make the following

Conjecture 2 If _Í`(a�` � c � `¾eÎ`Ïbdc�° }(²�c�z£³(aeÎ`(° , then ~ � X7�Y�[ G H ] is dense inSUT V W@XZYJ[ G\H^] .This problem is open even when

Yis a ball in h ¹ . We will prove below the equivalence of Conjectures 1

and 2.Parts of Theorem 4 were already known. Part a) is due to Escobedo [14]; so is part b), but in this case

the idea goes back to Schoen - Uhlenbeck [24] (see also Bourgain - Brezis - Mironescu [5]). For a�k � ,part c) is due to Schoen - Uhlenbeck [24]; their argument can be adapted to the general case (see, e.g.,Brezis - Nirenberg [12] or Brezis - Li [7]). The only new result is part d). The proof relies heavily on theComposition Theorem and Theorems 2 and 3. We do not know any direct proof of d). We also mentionthat for a�k � and

Y k4¿ H , Theorem 4 was established by Bethuel - Zheng [3]. For a general compactconnected manifold ¡ and for aJk � , the question of density of ~ � X �YJ[ ¡{] into

S H V W XjYJ[ ¡{] was settledby Bethuel [1] and Hang - Lin [15].

Remark 2 In Theorems 2 and 4, one may replaceY

by a manifold with or without boundary. The state-ments are unchanged. However, the argument in the proof of Theorem 1 does not quite go through to thecase of a manifold without boundary. Nevertheless, we make the following

Conjecture 3 LetY

be a manifold without boundary with Ð�ÑÓÒ Y }Ïz . ThenS T V W XjYJ[ ¡{] is path-

connected for every _K`¸aÔ`4bgc � `�ed`4b with a�eR`¨z , and for every compact connected manifold¡ .

Note that the condition Ð�ÑÓÒ Y }Cz is necessary, sinceS T V W X G H [ G H ] is not path-connected when aef} � .

Finally, we investigate the local path-connectedness ofS£T V W@XZYJ[ G\H^] . Our main result is

Theorem 5 Let _#`{aM`�bgc � `�eÕ`�b . ThenSUTLV W@XjYJ[ G\HE] is locally path-connected. Consequently,

the homotophy classes coincide with the connected components and they are open and closed.

The heart of the matter in the proof is the following

Claim. Let _#ÙaN`�bgc � `CeÕÙb . Then there is some Ö¦¤�_ such that, ifq¶q n ¬ � q¶q È É8Ê Ë `�Ö , then n

may be connected to � inS{TLV W

.As a consequence of Theorem 5, we have

Corollary 2 Let _B`ga�` � c � `"ef`db . Then´ nuµ T V W » mQn v �� [ � o S T V W XZYJ[ h·] x È É/Ê Ë kdn m v �¶� [ � o S T V W XjYJ[ h\] x È É/Ê Ë�tEquality in Corollary 2 follows from the well-known fact that

S�T V W|¯�× � is an algebra. The inclusionis a consequence of the fact that, clearly, we have´ n�µ T V W¸» m^n v �¶� [ � o S T V W XjYJ[ h·] xand of the closedness of the homotopy classes.

Another consequence of Theorem 5 is

Corollary 3 Conjecture 1 Ø Conjecture 2.

123


PROOF. By Corollary 2, we have´ nuµ T V W » n m v �� [ � o S T V W XjYJ[ h·] x È É/Ê Ë�tWe prove that the reverse inclusion follows from Conjecture 1. By Proposition 1 a) below, we may taken¦k � . Let �¦o ´ � µ T V W . By Theorem 5, there is some Ù-¤g_ such that

q¶q � ¬�Ú q¶q È É/Ê Ë `ÇÙ�Û Ú o ´ � µ T V W . LetX Ú=Ü ]¢ÝÞ~�� X �Y�[ G\H^] be such that Ú=Üf � inSUT V W

andqÓq Ú=Ü,¬ � qÓq È É/Ê Ë `dÙ . By Theorem 2 b), we obtain

that Ú=Ü and � are homotopic in ~�� X �YJ[ G\H^] . Thus Ú=Ü k v>�¶�wß for some globally defined smooth �lÜ . Hence�No m v �¶� [ � o S T V W XjYJ[ h·] x È É8Ê Ë�tConversely, assume that Conjecture 2 holds. Let nào S�TLV W@XjYJ[ G\HE] . By Theorem 2 a), there is

some Ú o{~�� X �YJ[ G\H^] such that Ú o ´ nuµ T V W . By Proposition 1 b), we have n �Ú o ´ � µ T V W . Thus n �Ú om v �� [ � o S T V W XZYJ[ h·] x È É/Ê Ë , so that clearly n �Ú o m v �� [ � o#~ � X �Y�[ hl] x È É/Ê Ë .

Finally, n#o m Ú v �� [ � o#~ � X �YJ[ hl] x È-É8Ê Ë , i.e. n may be approximated by smooth maps.

In the same vein, we raise the following

Open Problem 1. LetY

be a manifold with or without boundary. IsS£T V W@XZYJ[ ¡{] locally path-connected

for every a cje and every compact manifold ¡ ?The case a¢k � can be settled using the methods of Hang - Lin [15]. We will return to this question in a

subsequent work; see Brezis - Mironescu [11].The reader who is looking for more open problems may also consider the following

Open Problem 2. LetY ÝCh � be a smooth bounded domain. Assume _�`ga�`�bdc� `Ôef`gb and � ³�a�ef`gz (this is the range where ~�� X �YJ[ G\H^] is not dense in

S{T V W|XjYJ[ G\HE] ). Setá � kUm^nfo S T V W XZYJ[ G H ] [ n is smooth except at a finite number of points xwt(Here, the number and location of singular points is left free). Is

á � dense inS T V W XZYJ[ G H ] ?

Comment.á � is known to be dense in

S{T V W@XZYJ[ G\H^] in many cases, e.g.:a) a¢k � and � ³Ôe�`gz ; see Bethuel-Zheng [3]b) aJk � ¬ �>â e and z�`"ef`�² ; see Bethuel [2]c) a¢k �>â z and eMk:z ; see Riviere [20].

The paper is organized as follows

1. Introduction2. Proof of Theorem 13. Proof of Theorems 2 and 34. Proof of Theorem 45. Proof of Theorem 5Appendix A. An extension lemmaAppendix B. Good restrictionsAppendix C. Global liftingAppendix D. Filling a hole - the fractional caseAppendix E. Slicing with norm control

124


2. Proof of Theorem 1Case 1: a�ef` �When ae�` � , we have the following more general result

Theorem 6 If a"¤¸_�c � `Þe:`4bgcuaed` � and ¡ is a compact manifold, thenS�T V W@XZYJ[ ¡{] is path-

connected.

PROOF. Fix some r of¡ . For n#o S{T V W|XjYJ[ ¡{] , letãnMk¨ä nFc inYr c in hli,å Y t

Since ae£` � , we haveãn£o S T V Wæyç è X hli [ ¡{] . Let é Xjê c À ]Rk ãn X�À â X � ¬ ê ]�]ëc�_d³ ê ` � c À o Y

andé X � c À ]¦ì r . Then clearly é¸oO~ X ´ _�c � µ [ S{T V W�XjYJ[ ¡{]�] and é connects n to the constant r (here we useonly aef`Ç° ).

Case 2: � `gaef`Cz�c°±}CzIn this case one could adapt the tools developed in Brezis - Li [7], but we prefer a more direct approach.Let Ù¸¤í_ be such that the projection onto î Y be well-defined and smooth in the region m À ohli [

distX�À cî Y ]�`4z7Ù x . Let ïÍkàm À odh·i-å �YJ[

distX�À cî Y ]N`©Ù x . We have îuï(kíî YCðÔñ

, whereñ k±m À o�hli,å YJ[dist

XÁÀ cî Y ]-kgÙ x .Since � `�a�eÇ`{z , we have �>â eÕ`�a�` �=¥Þ�>â e ; thus, for nÇo S�T V W

we have tr nCo SUT �òH�Â W7V W . Letnfo SUT V W@XZYJ[ G\H^] . Fix some r o�GIH and define ��o SUT �2HÂ W7V W|X îuï [ GIH*] by

�Çk©ä tr nsc on îuïr c onñ t

We use the following extension result. (The first result of this kind is due to Hardt - Kinderlehrer - Lin[16]; it corresponds to our lemma when ófk � ¬ �>â e3c>ef`�z .)

Lemma 1 Let _Þ`�óô` � c � `ôe¨`õbgc�óöe¸` � . Then any �£o S�÷�V W=X îuï [ GIH*] has an extensionÚ o SÞ÷ � H�Â W7V W@X ï [ G\H^] .The proof is given in Appendix A; see Lemma A.1. It relies heavily on the lifting results in Bourgain -

Brezis - Mironescu [4].Returning to the proof of Case 2, with Ú given by Lemma 1, setãn¦kùøú û n in

YÚ in ïr in h Ü å XZYÔð ï�]Clearly,

ãnfo S TLV Wæyç è X hli [ GIHE] andãn is constant outside some compact set. As in the proof of Theorem 6,

we may useãn to connect n to r , since once more we have aef`�° .

Case 3: a�eMk � c=°±}�zThe idea is the same as in the previous case; however, there is an additional difficulty, since in the

limiting case aBk �>â e the trace theory is delicate - in particular, trS HÂ W7V W{ük ×�W

(unless eÔk � ). Insteadof trace, we work with a notion of “good restriction” developed in Appendix B; when a�k �>â z�c>e�kÞz , thespace of functions in ý HÂ � having _ as good restriction on the boundary coincides with the space ý HÂ �� ofLions - Magenes [17] (see Theorem 11.7, p. 72).

Our aim is to prove that any nfo S HÂ W>V W XjYJ[ G H ] can be connected to a constant r ofG H .Step 1: we connect n©o S H�Â W7V W@XZYJ[ G\H^] to some n H o S HÂ W>V W=XjYJ[ G\HE] having a good restriction

on î Y125


Let ÙÇ¤þ_ be such that the projection ÿ onto î Y be well-defined and smooth in the set m À oUhIi [dist

X�À c î Y ]N`gz>ÙD] x . For _B`�ÖB`ÕÙ , set ��ªkÞm À o YJ[dist

XÁÀ c î Y ]\kgÖ x . By Fubini, for a.e. _-`CÖB`�Ù ,we have n q �� o S HÂ W7V W X � � ] and

� � � � � q n X�À ] ¬ n X ] q Wq À ¬ òq i � H�� a Ä `�b t (1)

By Lemma B.5, this implies that n has a good restriction on � � , and that Rest n q � � kôn q � �a.e. on � � .

Let any _N`dÖ,`�Ù satisfying (1). For _N`� "`dÖ , let �� be the smooth inverse of ÿ q �� î Y .Let also

Y ��kºm À o YJ[dist

X�À cî Y ]¦¤� x . Consider a continuous family of diffeomorphisms � � � �Y Y � �7c_�³ ê ³ � , such that � � k id and � � q � � k�� . Thenê « n�®�� is a homotopy in

S HÂ W7V W . Moreover,if n � kôn�®�� , then n � kgn and n H q � � kdn q �� ®�� q � � . By (1), n H has a good restriction on î Y .

Step 2: we extend n H to h�iLet ï�kUm À o�hli,å �YJ[

distXÁÀ3[ î Y ]�`ÇÙ x . As in Case 2, we fix some r o#G�H and set

��k¨ä n H c on î Yr c onñ t

Clearly, �Mo S H�Â W7V W@X îuï�] , so that �¦o S�÷�V W@X îuï�] for _,`dó�` �>â e . We fix any _N`dó�` �7â e . By Lemma1, there is some Ú o S ÷ � HÂ W7V W X ï [ G H ] such that Ú q �� kd� . We define

ãn H k øú û n H c inYÚ c in ïr c in hli-å XjYOð ï�] t

We claim thatãn H o S H�Â W7V Wæyç è X h�i [ G\HE] . Obviously,

ãn:o S HÂ W>V Wæyç è X h·i,å Y ] . It remains to check thatãn H oS H�Â W7V W@XZYÔð ï�] . This is a consequence of

Lemma 2 Let _,`da�` � c � Òe#`Rbdc ae#} � and �M¤ga . Let n H o SUTLV W�XZY ] and Ú o S��EV W�X ï�] . Assumethat n H has a good restriction Rest n H q � � on î Y and that tr Ú q � � k Rest n H q � � . Then the map

ä n H c inYÚ c in ï

belongs toS{T V W@XZYOð ï�] .

Clearly, in the proof of Lemma 2 it suffices to consider the case of a flat boundary. WhenY kX ¬ � c � ]�iª�2H! X _�c � ] and ï¨k X ¬ � c � ]�i|�òH" X ¬ � c_ö] , the proof of Lemma 2 is presented in Appendix B;

see Lemma B.4.Returning to Case 3 and applying Lemma 2 with a:k �>â e3c#�©k ó ¥¨�7â e , we obtain that

ãn H oS H�Â W7V WæÓçè X hli-] . As in the two previous cases, this means that n H isS H�Â W7V W -homotopic to a constant.

Case 4: � ³gaef`Cz�c�°Îk �In this case,

Yis an interval. Recall the following result proved in Bourgain - Brezis - Mironescu [4]

(Theorem 1): ifY

is an interval and aef} � , then for each nfo S�T V W�XjYJ[ G\HE] there is some � o SUTLV W@XjYJ[ h\]such that nÔk v7�¶� . Recall also that, when aeK}Þ° , then ~�� X h [ hª] functions § with bounded derivativesoperate on

S{T V W; that is, the map �Õ« §,® � is continuous from

S�T V Winto itself (see, e.g., Peetre [19] foraeÕ¤{° , Runst - Sickel [23], Corollary 2 and Remark 5 in Section 5.3.7 or Brezis - Mironescu [9] whenaeNkR° ; this is also a consequence of the Composition Theorem). By combining these two results, we find

that the homotopyê « v7�Á� Hë�u�� connects nMk v7�¶� to � .

The proof of Theorem 1 is complete.

126


3. Proof of Theorems 2 and 3

We start with some useful remarks. For n#o S TLV W XZYJ[ G H ] , let´ nuµ T V W denote its homotopy class in

S T V W.

Proposition 1 Let _�`�a�`gbdc � `Ôe�`db . For nsc�No SUT V W=XjYJ[ G\HE] , we havea) n ´ � µ T V W k ´ n�� µ T V W ;b)

´ nuµ TLV W k ´ � µ T V W Ø ´ n ��Dµ TLV W k ´ � µ T V W ;c)

´ nuµ T V W ´ � µ T V W k ´ n�� µ T V W .

The proof relies on two well-known facts:S{TLV W=¯�× � is an algebra; moreover, if n Ü nFc�� Ü � inSUT V W

andqÓq n Ü qÓq $&% ³ô~Jc q¶q � Ü qÓq $&% ³Î~ , then n Ü � Ü n�� in

SUT V W. Here is, for example, the proof of c)

(using a)). Let first n H o ´ nuµ TLV W c�� H o ´ �Dµ T V W . If é@c(' are homotopies connecting n H to n and � H to � , thené�' connects n H � H to n�� ; thus´ n�µ T V W ´ �Dµ T V W Ý ´ nu�Dµ TLV W . Conversely, if Ú o ´ n�� µ T V W , then Ú o¦n ´ � µ T V W (by a)),

so that Ú �n"o ´ �Dµ T V W . Therefore, Ú k�n X Ú �n2]�o ´ n�µ TLV W ´ � µ T V W .We next recall the degree theory for

S{TLV Wmaps; see Brezis - Li - Mironescu - Nirenberg [8] for the

general case, White [25] when aMk � or Rubinstein - Sternberg [20] for the space ý H XjYJ[ G H ] andY

thesolid torus in h�¹ . Let _-`ga�`dbdc � `Ôef`db be such that aef}gz . Let n"o S{TLV W@X GIH) ñª[ G\HE] , where

ñis some open connected set in h+* . Clearly, for a.e. fo ñ c�n X-, c( �]Io S{T V W=X G\H [ G\H^] . For any such òcn X., c/ 0]is continuous, so that it has a winding number (degree) deg 0jn X., c/ 0].1 . The main result in [8] asserts that, ifae¦}gz , then this degree is constant a.e. and stable under

S£T V Wconvergence.

In the particular case where a�} � , there is a formula

degX n X-, c( �]],k �z32 � 4 � n X�À c( �]65 î�nî87 X�À c/ 0] � a Ä c

where n95M�Çkôn H � � ¬ n � � H . It then follows that, if a�} � and ae�}�z , we have

degX n q 4 ��:<; ]�k>= � ; = �4 � n X�À c/ 0]�5 îunî87 X�À c( �] � a Ä � t

Clearly, the above result extends to domains which are diffeomorphic to G=H6 ñ. In the sequel, we are

interested in the following particular case: let ? be a simple closed smooth curve inY

and, for small ÙB¤�_ ,let ?A@ be the Ù -tubular neighborhood of ? . We fix an orientation on ? .

Let � � G\HB O¿ @ ? @ be a diffeomorphism such that � q 4 � :&C �ED � G\HB �mQ_ x ? be an orientationpreserving diffeomorphism; here ¿ @ is the ball of radius Ù in h·iª�2H . Then we may define deg

X n q FHG ]Ôkdeg

X n�®�� q 4 � :&I G ] ; this integer is stable underS{T V W

convergence.We now prove b) of Theorem 2, which we restate as

Proposition 2 Let _"`{a�`£bdc � `�eÇ`{bdcuaeÕ}{z . Let nFc��"oC~�� X �YJ[ G\HE] . Then´ n�µ T V W k ´ �Dµ T V W if

and only if n and � are ~�� - homotopic.

PROOF. Using Proposition 1, we may assume �#k � . Suppose first that nÕoÕ~B� X �YJ[ GIHE] and � are ~J� -homotopic. Then n and � are

S�T V W-homotopic. Indeed, when aÔk � , this is proved in Brezis - Li [7],

Proposition A.1; however, their proof works without modification for any a . We sketch an alternative proof:since n and � are ~ � -homotopic, there is some � od~ � X �YJ[ hl] such that nÇk v �¶� . Then

ê « v �Q� Hë�u�� connects n to � in

SUT V W.

Conversely, assume that the smooth map n isS�T V W

-homotopic to � . By continuity of the degree, wethen have deg

X n q F G ],kÎ_ for each ? . Since n is smooth, we obtain_�k degX n q F G ]-k deg

X n q F ]�k �zJ2 � F n95 îunî87 � a t127


Thus the closed form K¾kgnL5NM,n has the property that O F K , 7 � a�kR_ for any simple closed smooth curve? . By the general form of the Poincare lemma, there is some � of~ � X �Y�[ h·] such that KàkPM � . One mayeasily check that nMk v7��¶�w�RQ � for some constant ~ . Then

ê « v7�� Hë�u�� ¶�w�RQ � connects n to � in ~�� X �YJ[ G\H^] .We now turn to the proof of the remaining assertions in Theorems 2 and 3.

Case 1: a�ef}C°fc�°±}gzStep 1: each nfo S{T V W@XZYJ[ G\H^] can be connected to a smooth map ��of~�� X �Y�[ G\H*]This is proved in Brezis - Li [7], Proposition A.2, for aBk � and eK}�° ; their arguments apply to anya and any e such that aeC}�° . The main idea originates in the paper Schoen - Uhlenbeck [23]; see also

Brezis - Nirenberg [12], [13].Step 2: we have

´ n�µ TLV W k±m^n v>�¶� [ � o SUT V W|XjYJ[ h·] xLet � o SUT V W|XjYJ[ h·] . Then

ê « ¬u n v>�Á� HL�� connects n v7�¶� to n inS{T V W

. (Recall that, if §"of~�� X h [ hª]has bounded derivatives and ae�}:° , then the map �C« §N® � is continuous from

S£TLV Winto itself.) This

proves “ » ”. To prove the reverse inclusion, by Proposition 1, it suffices to show that´ � µ T V W Ý m v �� [ � oSUT V W@XZYJ[ h·] x .

Let �Ço ´ � µ T V W . For eachÀ o Y

, let ¿ Ä Ý Ybe a ball containing

À. We recall the following lifting

result from Bourgain - Brezis - Mironescu [4] (Theorem 2): if é is simply connected in h�i and aeO}R° ,then for each Ú o S{T V W@X é [ GIHE] there is some S�o S{T V W@X é [ h·] such that Ú k v>�UT . Thus, for eachÀ o Y

there is some � Ä o SUT V W@X ¿ Ä [ h·] such that � q I�V k v>�� V . Note that , in ¿ Ä ¯ ¿�W , we have� Ä ¬K� W o SUTLV W@X ¿ Ä ¯ ¿ W [ z32RX¢] . Therefore, � Ä ¬�� W oY'�¡�Z X ¿ Ä ¯ ¿ W [ z32RX¢] , since aeO}R° . It thenfollows that � Ä ¬�� W is constant a.e. on ¿ Ä ¯ ¿ W ; see Brezis - Nirenberg [12], Section I.5.

By a standard continuation argument, we may thus define a (multi-valued) argument � for � in thefollowing way: fix some

À � o Y. For any

À o Y, let [ be a simple smooth path from

À � toÀ

. Then,for ÙÇ¤Î_ sufficiently small, there is a unique function �]\ o SUTLV W@X [ @ [ hl] such that � q \ G k v>�¶�_^ and�`\ q I G � Äba�� k � Ä�a q I G � Äba�� ; here, [c@ is the Ù -tubular neighborhood of [ . We then set� q I G � ÄQ� k � \ q I G � ÄQ� t

We actually claim that � is single-valued. This follows from

Lemma 3 Assume that _�`�a,`�bdc � `CeO`{bdcuaeO}Þ°fc�° }Þz . If Ú o S{T V W@X G\H� #¿ H [ G\HE] is suchthat deg

X Ú q 4 � :&I � ]lkR_ , then there is some Sgo S{T V W=X G\H� �¿ H ] such that Ú k v>�UT .

Here, ¿ H is the unit ball in h·iª�2H . The proof of Lemma 3 is presented in Appendix C; see Lemma C.1.Returning to the claim that � is single-valued, we have that deg

X � q FHG ]-k _ for each ? , since ��o ´ � µ TLV W .By Lemma 3, a standard argument implies that � is single-valued.

The proof of Theorems 2 and 3 when a�ef}C° is complete.

Case 2: a�} � c � `Ôef`gbgcu°Ì}�²�c�zB³gaef`Ç°Step 1: we have

´ nuµ T V W kUm^n v>�¶� [ � o SUT V W|XjYJ[ hl] ¯,S H V TZW@XjYJ[ h·] x For “ » ”, we use the Composition

Theorem mentioned in the Introduction, which implies thatê « n v �Á� HL�� connects n v>�� to n in

SUTLV W.

For “ Ý ” it suffices to prove that´ � µ TLV W Ý m vQ�� [ � o SUTLV W@XjYJ[ h·] ¯�S H V TjW|XjYJ[ h·] x . We proceed as

in Case 1, Step 2. Let ��o ´ � µ T V W . The corresponding lifting result we use is the following (see Bourgain- Brezis - Mironescu [4], Lemma 4): if aÞ} � c�aeô}(z and é is simply connected in hIi , then foreach Ú o SUT V W|X é [ G\H*] there is some S�o SUTLV W=X é [ h·] ¯ÕS H V TjW@X é [ h·] such that Ú k v>�UT . As inCase 1, for each

Àthere is some � Ä o SUT V W@X ¿ Ä [ h·] ¯KS H V TjW@X ¿ Ä [ h·] such that � q I�V k v>�¶� V . Since� Ä ¬¸� Wfo S H V H X ¿ Ä ¯ ¿�W [ z32RX¢] , we find that � Ä ¬Õ� W is constant ae. on ¿ Ä ¯ ¿�W (see [4], Theorem

B.1.). These two ingredients allow the construction of a multi-valued phase � o SÍT V W¢¯#S H V TjW for � . Toprove that � is actually single-valued, we rely on

Lemma 4 Assume that aN} � c � `ÇeK`{bdc�° }Þ²�c�z¦³�aeK`�° . If Ú o S{T V W@X G\HY U¿ H [ G\HE] is suchthat deg

X Ú q 4 � :dI � ]¦k±_ , then there is some S�o S{T V W@X G\He f¿ H [ h·] ¯fS H V TjW=X G\HY �¿ H [ h·] such that�Çk v �fT .

128


The proof of Lemma 4 is given in Appendix C; see Lemma C.2.The proof of Step 1 is complete.Step 2: assume ad} � c � `þe¨`ºbgc�a�e¨}ºz ; then, for each n¸o S{TLV W=XZYJ[ G\H^] , there is some�No SUT V W@XZYJ[ G\HQ] ¯ ~�� XZYJ[ G\H^] such that �No ´ n�µ TLV W .Consider the form K kông5�M,n . Then K±o S{T �2H V W|XjY ] ¯�×�TjW@XZY ] (see Bourgain - Brezis - Mironescu

[4], Lemmas D.1 and D.2). Let � o S�T V W|XjYJ[ h·] ¯#S H V TjW@XZYJ[ h·] be any solution of h � k div K inY

.By the Composition Theorem, we then have v � �� o SUT V W=XjYJ[ G\HE] , and thus �dkÌn v � �¶� o SUT V W@XZYJ[ G\H^] .We claim that �Kog~�� XjYJ[ G\HE] . Indeed, let ¿ be any ball in

Y. Since af} � and aeC}Íz , there is someSgo SUTLV W@X ¿ [ h\] ¯�S H V TjW=X ¿ [ h\] such that n q I k v>�UT . It then follows that K q I kiMNS . Thus h � kjh9S

in ¿ , i.e., S ¬,� is harmonic in ¿ . Since in ¿ we have ��kôn v � �¶� k v �Á�fT � � � , we obtain that �No#~ � X ¿-] ,so that the claim follows.

Using Step 1 and the equality �-kgn v � �� , we obtain that �No ´ n�µ T V W .Step 3: for each nfo SUT V W=XjYJ[ G\HE] , there is some Ú o#~�� X �YJ[ GIHE] such that Ú o ´ n�µ TLV W .In view of Step 2, it suffices to consider the case where n�o S�T V W=XjYJ[ GIHE] ¯ ~�� XjYJ[ G\HQ] . We use the

same homotopy as in Step 1, Case 3, in the proof of Theorem 1:ê « n¦®6� � , where � � is a continuous

family of diffeomorphisms � � � �Y Y � � such that � � kPk � . Clearly, �Çkôn�®�� H o�~�� X �Y�[ G\H^] .The conclusions of Theorems 2 and 3 when a�} � c � `ReC`©bdc�° }�²�cuz"³£aeC`£° follow from

Proposition 2 and Steps 1 and 3.We now complete the proof of Theorem 2 with

Case 3: _B`ga�` � c � Òef`gbdc�°±}C²�c�z�³Caef`�°In this case, all we have to prove is that, for each ngo S�T V W=XjYJ[ G\HE] , there is some ��og~�� X �Y�[ GIHE]

such that ��o ´ n�µ TLV W . The ideas we use in the proof are essentially due to Brezis - Li [7] (see l 1.3, “Filling”a hole).

We may assume that n is defined in a neighborhood m of�Y

; this is done by extending n by reflectionsacross the boundary of

Y- the extended map is still in

S�T V Wsince _�` aK` � . We next define a good

covering ofY

: let Ù�¤Ç_ be small enough; forÀ o¦h\i , we setn Äi kjo©m À ¥ Ù3p ¥ X _�c/ÙD] i [ pso"X i and

À ¥ ÙJp ¥ X _�c/ÙD] i ÝPm xDtDefine also

n Ä� crq�k � c t¶tÓt c° ¬ � , by backward induction :n Ä� is the union of faces of cubes in

n Ä� � H .By Fubini, for a.e.

À oCh·i , we have n q s V� o SUT V W c_q"k � c t¶tÓt c�° ¬ � , in the following sense: since�>â e#`RaB` � , we have tr n q s Vtvu � o SUT �2HÂ W7V W for allÀ

. However, for a.e.À

, we have the better property trn q s Vtvu � k n q s Vtvu � o SUTLV W. For any such

À, we have

tr w^n q s Vtvu �/xzyy s Vtvu � o SUT �òH�Â W7V W , but once more for a.e. suchÀ

we have the better property tr w^n q s Vtvu �/xzyy s Vtvu �k�n q s Vtvu � o S T V W, and so on. (See Appendix E for a detailed discussion).

We fix anyÀ

having the above property and we drop from now on the superscriptÀ

.Step 1: we connect n to some smoother map n H Let {Þk ´ ae�µ , so that zR³|{{³±° ¬ � . Sincen q s�} o SUT V W

and a�ef}~{ , there is a neighborhood ï ofn * in

n * � H and an extensionãn"o S{T � H�Â W7V W|X ï [ GIHE]

of n q s�}. This extension is first obtained in each cube ~ÍÝ n * � H starting from n q � Q (see Brezis - Nirenberg

[12], Appendix 3, for the existence of such an extension). We next glue together all these extensions toobtain

ãn [ ãn belongs toS{T � HÂ W>V W since �7â e�Ùa ¥:�7â e�` �@¥:�>â e . Moreover, the explicit construction in

[12] yields someãn�oÇ~�� X ï�å n * ] . We next extend

ãn ton * � H in the following way: for each ~ÎÝ n * � H ,

let � Q be a convex smooth hypersurface in ~ ¯ ï . Since � Q is { -dimensional and {Õ}£z�c ãn q ��may be

extended smoothly in the interior of � Q as an GIH -valued map (here, we use the fact that 2 * X GIHE]-k _ ). Letãn Q be such an extension. Then the map��k ä ãnFc outside the � Q ’sãn Q c inside � Qbelongs to

S{T � HÂ W7V W=X n * � H ] . To summarize, we have found some �Uo S�T � HÂ W7V W=X n * � H [ G\HE] such that� q s�} kgn q s�}.

129


Pick any aC`Ìa H ` min m>a ¥:�>â e3c � x and let e H be such that a H e H k{ae ¥:� (note that � `Õe H `Þb ).By Gagliardo - Nirenberg (see, e.g., Runst [22], Lemma 1, p.329 or Brezis - Mironescu [10], Corollary 3),we have

SUT � H�Â W7V W=¯¦× �4Ý SUT � V W � . Thus �No S{T � V W � X n * � H ] .We complete the construction of the smoother map n H in the following way: if {"kÍ° ¬ � , then � is

defined inn i and we set n H kg� ; if {M`C° ¬ � , we extend � to

n i with the help of

Lemma 5 Let _Í` a H `Ïbdc � `àe H `Ïbdc � ` a H e H ` °fc ´ a H e H µ"³�q¨`(° . Then any �¸oS T � V W � X n � [ G H ] has an extension n H o S T � V W � X n i [ G H ] such that n H q s�� o S T � V W � for pòk�q c tÓt¶t c° ¬ � .

When a H k � , Lemma 5 is due to Brezis - Li [7], Section 1.3, “Filling” a hole; for the general case, seeLemma D.3 in Appendix D.

We summarize what we have done so far: if {Ík ´ ae�µ , then there are some a H cje H such that a:à H ` � c � `ge H `£bgc�a H e H k©a�e ¥Þ� and a map n H o SUT � V W � X n i [ GIH*] such that n H q s � o SUT � V W � ccq�k{0c t¶t¶t c ° ¬ � and n H q s�} k�n q s�}. By Gagliardo - Nirenberg and the Sobolev embeddings, we have in particularn H q s � o SUT V W c_qBk�{0c t¶t¶t c ° ¬ � . Finally, n and n H are

SUT V W- homotopic by

Lemma 6 Let _�`:a�` � c � `�eÔ`�bgc � `:aeO`R°fc ´ ae�µl³�q�`:° . If n q s�� o SUT V W c�n H q s�� o SUT V W c�pFkqDc t¶tÓt c° , and n q s � kgn H q s � , then n and n H areS T V W

-homotopic.

The case a-k � is due to Brezis - Li [7]; the proof of Lemma 6 in the general case is presented in theAppendix D- see Lemma D.4.

Step 2: induction on´ ae�µ .

If {Rk ´ ae�µ�k±° ¬ � , we have connected in the previous step n to n H o SUT � V W � X n i [ G\H^] , wherea�`ga H ` � c � `Ôe H `gb and a H e H k:a�e ¥g� }�° . Using Case 1 (i.e., aef}�° ) from this section, n H maybe connected in

S{T � V W � (and thus inS{T V W

, by Gagliardo - Nirenberg and the Sobolev embeddings) to some�Nof~�� X �YJ[ GIHE] . This case is complete.If {Nk ´ ae�µ2k:° ¬ z , then

´ a H e H µòk:° ¬ � . By the previous case, n H can be connected inS{T � V W � (and

thus inS{T V W

) to some �No#~�� X �Y�[ G\HE] . Clearly, the general case follows by induction.The proof of Theorems 2 and 3 is complete.We end this section with two simple consequences of the above proofs; these results supplement the

description of the homotopy classes.

Corollary 4 Let _�`õag` bgc � `þe¸`õbdcuae¨}ºz�c�° }�z . For nFc��o S{T V W@XZYJ[ G\H^] , we have´ nuµ T V W k ´ �Dµ T V W Ø degX n q F G ]·k deg

X � q F G ] for every ? .

Corollary 5 Let _Í` a H c a � ` bgc � `àe H cje � `Ïbdc�a H e H } z�c�a � e � } z�c�° }Ïz . For nFc��¸oS T � V W � XZYJ[ G H ] ¯¦S T � V W � XjYJ[ G H ] , we have´ nuµ T � V W � k ´ �Dµ T � V W � Ø ´ nuµ T � V W � k ´ � µ T � V W � .

Clearly, Corollary 5 follows from Corollary 4. As for Corollary 4, let n H c�� H o#~�� X>�YJ[ G\HE] be such that´ n H µ T V W k ´ n�µ T V W and´ � H µ T V W k ´ �Dµ TLV W . Then, by Theorem 2 b),´ nuµ T V W k ´ �Dµ TLV W Ø ´ n H µ TLV W k ´ � H µ T V W Ø ´ n H µ Q a k ´ � H µ Q a Ø deg

X n H q F ]-k degX � H q F ]Lc��? t (2)

Moreover, we have

degX n H q F ]·k deg

X � H q F ]FØ degX n H q FHG ]lk deg

X � H q FHG ]�Ø degX n q F_G ]·k deg

X � q FHG ]ëc#��?\c (3)

by standard properties of the degree.We obtain Corollary 4 by combining (2) and (3).

130


4. Proof of Theorem 4

According to the discussion in the Introduction, we only have to prove part d). Let aC} � c � `¸e£`bgc�° }�²�c�zM³Ua�eK`Þ° . Let nKo SUTLV W=XZYJ[ G\H^] . By Theorem 2 a), there is some �#oK~�� X �YJ[ G\H^] suchthat ��o ´ nuµ T V W . By Theorem 3 b), there is some � o S{T V W=XjYJ[ h\] ¯fS H V TjW@XZYJ[ h·] such that �Nk:n v>�� . LetX ��Ü ]�Ý�~�� X �YJ[ h·] be such that ��Ü,�� in

SUT V W@¯MS H V TjW . By the Composition Theorem, the sequence ofsmooth maps

X � v � ��wß ] converges to n inS{T V W=XjYJ[ G\HE] . The proof of Theorem 4 is complete.

5. Proof of Theorem 5

We start this section with a discussion on the stability of the degree: recall that if aef}Cz , then degX n q F G ] is

well-defined and stable underS{T V W

convergence. However, while the condition a�eÕ}�z is optimal for theexistence of the degree (see Brezis - Li - Mironescu - Nirenberg [8], Remark 1), the stability of the degreeof

S T V Wmaps holds under (the weaker assumption of)

S T � V W � convergence, where a H e H } � . This propertyand Corollary 4 suggest the following generalization of Theorem 5

Theorem 7 Let _#`�aM`£bdc � `geÇ`£bgcu_#`�a H `�aDc � `ge H `£bgc � ³�a H e H ³�ae . Then for eachnfo SUT V W@XZYJ[ G\H^] there is some Ö�¤C_ such thatm^�No S T V W XjYJ[ G H ] [>qÓq � ¬ n q¶q È É � Ê Ë � `�Ö x Ý ´ n�µ T V W tNote that

SUT V W|XjYJ[ G\HE]MÝ SUT � V W � XZYJ[ G\H^] , by Gagliardo - Nirenberg and the Sobolev embeddings, sothat Theorem 5 follows from Theorem 7 when aef}Cz (when a�e�`gz , there is nothing to prove, by Theorem1).

Proof of Theorem 7Step 1: reduction to special values of aDc a H cÁescÁe H .We claim that it suffices to prove Theorem 7 when_-`Ca H `�a�` � ¬ X ° ¬ � ] â esc � `"ef`dbgc � Òe H `dbgc�aeMk:z�c�a H e H k � c°Ì}�z t (4)

Indeed, assume Theorem 7 proved for all the values of aDc a H cÁescÁe H satisfying (4). Let _Ô`Ía � `©bdc � è � `Ìbgc ° }àz be such that a � e � }àz (when ° k � or a � e � `Ìz , there is nothing to prove). LetnÕo SUT a V W a and let aDc a H cÁescÁe H satisfy (4) and the additional condition aM`{a � . By Gagliardo - Nirenbergand the Sobolev embeddings, there is some Ö � ¤�_ such that¡ k m^��o SUT a V W a XjYJ[ G\HE] [Qq¶q � ¬ n qÓq È É a Ê Ë a `CÖ � x Ým^��o SUTLV W@XjYJ[ G\HE] [Qq¶q � ¬ n qÓq È É � Ê Ë � `CÖ xwt (5)

By the special case of Theorem 7, we have �Mo#¡ Ûº��o ´ nuµ T V W . By Corollary 5, we obtain ¡ Ý ´ n�µ T a V W a ,i.e.,

´ n�µ T a V W a is open.In conclusion, it suffices to prove Theorem 7 under assumption (4). Moreover, by Proposition 1 we may

assume nMk � .Step 2: construction of a good covering.We fix a small neighborhood m of

�Y. By reflections across the boundary of

Y, we may associate to eachnfo SUT V W@XZYJ[ G\H^] an extension

ãnfo SUT V W�X m [ GIHE] satisfyingqÓq ãn ¬ ã� q¶q È É8Ê Ë �� ³�~ H q¶q n ¬ � qÓq È É/Ê Ë �� (6)

and q¶q ãn ¬ ã� qÓq È É � Ê Ë � �� ³�~ H q¶q n ¬ � qÓq È É � Ê Ë � �� t (7)

131


In this section, ~ H cL~ � c tÓt¶t denote constants independent of nFc��c t¶t¶t .We fix some small ÙO¤£_ . By Lemma E.2 in Appendix E, for each ��o S£T V W=XjYJ[ GIHE] there is someÀ o¦h i (depending possibly on � ) such that the covering

n Äi has the properties� q s V� o S T V W c+qBk � c tÓt¶t c ° ¬ � (8)

and qÓq � q s V� ¬ � q¶q È É � Ê Ë � � s V� � ³g~I� qÓq � ¬ � q¶q È É � Ê Ë � �� ³�~I�>~ H q¶q � ¬ � qÓq È É � Ê Ë � �� (9)

(the last inequality follows from (7)).While

Àmay depend on � , the covering

n Äi has two features independent of � :

the number of squares inn Ä� has a uniform upper bound � [

(10)

if ~�H7c ~ � are two squares inn Ä� , there is a path of squares in

n Ä� each onehaving an edge in common with its neighbours, connecting ~�H to ~ � . (11)

Step 3: choice of Ö .We rely on

Lemma 7 Let ~{k X _�c/ÙD] � and _,`Ra H ` � c � `Ke H `:bdcua H e H k � t Then for each Ö H ¤d_ there is someÖE��¤C_ such that every map ��o S{T � V W � X î2~ [ G\H^] satisfyingqÓq � ¬ � q¶q È É � Ê Ë � � � Q � `CÖ^� (12)

has a lifting � o SUT � V W � X î2~ [ hl] such thatqÓq � qÓq È É � Ê Ë � � � Q � `CÖ H t (13)

Clearly, in Lemma 7, ~ may be replaced by the unit disc. For the unit disc, the proof of Lemma 7 isgiven in Appendix C; see Lemma C.3. In particular, if (12) holds, then we haveqÓq � qÓq $ � � � Q � `C~ ¹ Ö H (14)

for some ~ ¹ independent of the Ö3� s. We now take Ö H such thatÖ H `�20Ù â ~ ¹ t (15)

With Ö � provided by Lemma 7, we chooseÖ�k min m>ÖE� â ~ � c ÖE� â ~ H ~�� xwt (16)

Step 4: construction of a global lifting for � q s V� .Let ��o SUT V W=XjYJ[ G\HE] satisfy

q¶q � ¬ � qÓq È É � Ê Ë � `�Ö . Since ÖB³gÖE� â ~ H ~�� , (9) implies that the conclusionof Lemma 7 holds for � q � Q and every square ~ in

n Ä� . Thus, for every ~õo n Ä� , � q � Q has a lifting � Qsatisfying (14) and � Q o SUT � V W � X î2~�] .We claim that � Q o SUT V W=X î2~�] . The statement being local, it suffices to prove that � Q o SUTLV W@XÁ× ] ,

where×

is the union of three edges in î2~ . Since×

is Lipschitz homeomorphic with an interval, by Theorem1 in [4] there is some Sgo S{TLV W@XÁ× ] such that ��k v>�UT in

×(here we use _-`Ca�` � and aeMk:z�} � ). In

×,

we have S ¬f� Q o XZSUT V W ¥ SUT � V W � ] XÁ×|[ zJ2RX¢] ; thus S ¬#� Q is constant a.e. in×

(see [4], Remark B.3), sothat the claim follows.

Since aef¤ � and � q s V� o SUT V W c � Q o SUTLV W, we may redefine � q s V� and � Q on null sets in order to have

continuous functions. We claim that the function � X ]Ik � Q X ] , if oO~ is well-defined on

n ÄH (and thus

132


continuous andS{T V W

). By (11), it suffices to prove that, if ~BH>cL~ � are squares inn Ä� having the edge � in

common, then � Q � k � Q � on � . Clearly, on � we have � Q � k � Q � ¥ z3p2 for some pFo"X . Thusq¶q � Q � ¥ zHp2 q¶q $ � �f� � k qÓq � Q � q¶q $ � �U� � `g~ ¹ Ö H cby (14). It follows that z q p q 20Ù�k qÓq z3p�2 qÓq $ � �f� � ³ q¶q � Q � q¶q $ � �f� � ¥ ~ ¹ Ö H `gz ~ ¹ Ö H c (17)

which implies p3kd_ by (15) and (16).In conclusion, � q s V� has a global lifting � o S T V W X n ÄH [ h·] .Step 5: construction of a good extension Ú of � q s V� .Let � � o SUT � H�Â W7V W=X n Ä� [ h·] be an extension of � , � ¹ o SUT � � Â W7V W@X n Ä¹ [ h·] an extension of � � , and

so on; let � i o SUT �� iª�2H��ÁÂ W7V W@X n Äi [ h·] be the final extension. Note that these extensions exist sinceaÔ` �¢¥ X ° ¬ � ] â e , so that trace theory applies. We set Ú k v7�� t o SUT �l� iª�2H��ÁÂ W>V W=X n Äi [ G\H^] . SinceX a ¥ X ° ¬ � ] â e2] , eMkd° ¥�� ¤�° , we obtain by Theorem 3 that Ú o ´ � µ T �� i|�òH/�jÂ W7V W . By Corollary 5, wealso have Ú o ´ � µ T V W .

We complete the proof of Theorem 7 by provingStep 6: Ú o ´ �Dµ T V W .We rely on the following variant of Lemma 6

Lemma 8 Let _�`Ra-` � c � `�eÔ`�bgc � `:aeO`R°fc ´ a�e�µ·³�q¦`R° . Let ��c Ú o S{T V W@X n i [ GIHE] be suchthat � q s�� o SUT V W c Ú q s�� o SUT V W c�psk�q c tÓt¶t c° ¬ � . Assume that � q s � and Ú q s � are

SUT V W-homotopic. Then �

and Ú areSUT V W

-homotopic.

The proof of Lemma 8 is given Appendix D; see Lemma D.5.When °í}£² , we are going to apply Lemma 8 with qÔk¸z . In order to prove that � q s � and Ú q s � areSUT V W

-homotopic, it suffices to find, for each ~(o n � , a homotopy é Q from � q Q to Ú q Q preserving theboundary condition on î2~ ; we next glue together these homotopies (this works since _�`þaO` � ). Weconstruct é Q using the lifting: since ae�kRz�k dim ~ and ~ is simply connected, by Theorem 2 in [4] thereis some Sgo S{T V W@X ~ [ h·] such that ��k v7�fT in ~ . By taking traces, we find that � q � Q k v>� tr T k v>��

�; thus

tr S ¬�� Q o XjSUT �2HÂ W7V W ¥ SUT V W ] X î2~ [ z32RXJ] . Therefore, tr S ¬�� Q is constant a.e., by Remark B.3 in [4].We may assume that tr SÇk � Q k tr � � . Then

ê « ¬u v �� HL�� T�� is the desired homotopy é Q .When °¾k:z , the above argument proves directly (i.e., without the help of Lemma 8) that Ú o ´ � µ T V W .The proof of Theorem 7 is complete.

Appendix A An extension lemma

In this appendix, we investigate, in a special case, the question whether a map inS�÷ V W@X î�ï [ G\H*] admits an

extension inSÞ÷ � H�Â W7V W@X ï [ G\H^] .

Lemma A.1 Let _N`dóK` � c � `KeÔ`:bgc�óöeO` � c�°õ}Rz . Let ï be a smooth bounded domain in h\i .Then every ��o S�÷�V W=X îuï [ G\H^] has an extension Ú o S�÷ � H�Â W7V W=X ï [ G\H^] .PROOF. We distinguish two cases: ó"³ � ¬ �7â e and óÔ¤ � ¬ �7â e .Case ód³ � ¬ �>â e : since óöeC` � c�� may be lifted in

S�÷�V W(see Bourgain - Brezis - Mironescu [4]), i.e.

there is some SRo S�÷�V W@X îuï [ h·] such that �Nk v>�UT . Let � o SÞ÷ � HÂ W7V W=X ï [ h·] be an extension of S . ThenÚ k v>�� o SÞ÷ � H�Â W7V W=X ï [ G\H^] (since ó ¥g�7â e�³ � andÀ « v>� Ä is Lipchitz). Clearly, Ú has all the required

properties.Case óK¤ � ¬ �7â e : the argument is similar, but somewhat more involved. The proof in [4] actually yieldsa lifting which is better than

S�÷ V W; more specifically, this lifting S belongs to

S � ÷�V W Â� for _¦` ê ³ � , seeRemark 2, p.41, in the above reference. On the other hand, since ó"¤ � ¬ �>â e , we have

ê k�e â X óöe ¥#� ]I` � .

133


For this choice ofê, we obtain that � has a lifting Sgo SU÷�V W�¯MS HL�òH�Â � ÷QW � H�� V ÷QW � H . This S has an extension� o SÞ÷ � HÂ W7V W�¯�S H V ÷QW � H . By the Composition Theorem stated in the Introduction, the map Ú k v ��

belongs toS�÷ � H�Â W7V W|X ï [ GIH*] . Clearly, we have tr Ú kd� .

Remark A.1. The special case eÔ`Rz and ó�k � ¬ �>â e was originally treated by Hardt - Kinderlehrer -Lin [16] via a totally different method. Their argument extends to the case ef`dz and óöe#` � , but does notseem to apply when ef}gz .

Appendix B Good restrictions

In this appendix, we describe a natural substitute for the trace theory when aNk �7â e ; it is known that thestandard trace theory is not defined in this limiting case.

For simplicity, we consider mainly the case of a flat boundary. However, we state Lemma B.5 (used inthe proof of Theorem 1) for a general domain. We start by introducing someNotations: let �Uk X _�c � ]�iª�2HQc Y � k�� X _�c � ]ëc Y � k�� X ¬ � c_ö]Lc Y k Y � ðMY � k�� X ¬ � c � ] .If � is a function defined on � , we set

ã� X�À �Pc ê ]-kô� XÁÀ ] forXÁÀ �Zc ê ]Io Y

.

Lemma B.1 Let _�`ga�` � c � Òef`gb . Then for nfo S{T V W=XjY � ] and for any function � defined on � ,the following assertions are equivalent:a) �Mo S{T V W@X ��] and � k �� q n X�À ] ¬ ã� X�À ] q WÀ TjWi � À `db [ X�� t¶� ]b) the map Ú H k ä nFc in

Y �ã��c inY � c belongs to

S{T V W=XjY ] ;c) the map Ú �ªk©ä n ¬ ã��c in

Y �_�c inY � belongs to

S{T V W@XZY ] .PROOF. Recall that, if é is a smooth or cube-like domain, then an equivalent (semi-) norm on

S©T V W=X é�]is given by

§ « ¬� �� i�� ½ H� �� C ÄH�c�R�ZÄ � � � �c�AD § X�À ¥ ê v �Q] ¬ § X�À ] q Wê TjW � H � À � ê¢¡E£¤ HÂ W X�� t zD]

(see, e.g., Triebel [25]).Clearly, both b) and c) imply that �Ôo S{T V W=X ��] . Conversely, for �"o S{T V W=X ��] we have to prove the

equivalence of (B.1), b) and c). We consider the norm given by (B.2). Taking into account the fact thatÚ H c Ú � belong toSUTLV W

inY � and

Y � , we see thatÚ H o S T V W XZY ]�Ø¦¥fk � �� òH q n X�À ] ¬ ã� XÁÀ ] q WX�À i ¬ ê ] TjW � H§� ê � À `db X�� t ²w]and Ú � o S TLV W XjY ]BØ¦¥O`gb t X�� t ¨ ]The lemma follows from the obvious inequality� ¬ zö� TZWa�e � ³©¥�³ �ae � t

We now assume in addition that aef} � and derive the following

Corollary B.1 Let _-`Ca�` � c � `Ôef`gb be such that ae�} � . Then, for every n#o S TLV W XZY � ] we have

134


a) for each _-³ ê � ` � , there is at most one function � defined on � such that the mapsÚ � aH k¨ä nsc in �ª XÁê � c � ]ã�uc in �ª X ¬ � c ê � ]and Ú �a� k¨ä n ¬ ã�0c in �« Xjê � c � ]_�c in �« X ¬ � c ê � ]belong to

S T V W XjY ] ;b) for a.e. _-³ ê � ` � , the function ��kdn X-, c ê � ] has the property that Ú �aH c Ú �a� o S T V W XZY ] .(As usual, the uniqueness of � is understood a.e.)

The above corollary suggests the following

Definition: let _-`Ca�` � c � Òe�`gbdcuaef} � c�_-³ ê � ` � . Let nfo SUT V W=XjY � ] and let � be a functiondefined on � . Then � is the downward good restriction of n to m À i k ê � x if Ú � aH c Ú � a� o SUT V W|XjY ] ; wethen write �"k Rest n q �Ä t ½ �a . Similarly, for _"` ê � ` � we may define an upward good restriction Restn q �Ä t ½ �a kg� as the unique function � defined on � satisfying the two equivalent conditions

a)S �aH k©ä ã��c in �ª XÁê � c � ]nFc in �ª X _�c ê � ] o S TLV W XjY � ]

and

b)S �a� k©ä _�c in �« Xjê � c � ]n ¬ ã��c in �« X _�c ê � ] o S T V W XjY � ] t

If � is both an upward and a downward good restriction, we call it a good restriction and we write �OkRest n q Ä t ½ �a .Corollary B.2 Let _,`daB` � c � `�e#`Rbgc�a�e"} � . Let n"o S{T V W@XZY � ] . Then, for a.e. _,` ê � ` � , wehave Rest n q Ä t ½ �a kdn X-, c ê � ] .Remark B.1 If aeÍ¤ � , then functions n©o S�T V W@XZY � ] have traces for all _d³ ê � ³ � . However,these traces need not be good restrictions. Here is an example: For °Ïk¸z , one may prove that the mapÀ « X�À ¬ �7â z v H ] â q À ¬ �>â z v H q belongs to

SUT V W=XjY ] if _-`�a�` � c� `Ôef`gbgcLa�ef`gz . However, if a�ef¤ � , its trace

tr n q Ä � ½ � k¨ä � c ifÀ H ¤ �>â z¬ � c ifÀ H ` �>â z

does not belong toS{T V W|X _�c � ] , so that it is not a good restriction.

Remark B.2 In the limiting case a�k �7â e , functions inS�T V W

do not have traces. However, they do havegood restrictions a.e.

Here is yet another simple consequence of Lemma B.1

Corollary B.3 Let _�`�a�` � c � `Ôef`dbgc�aef} � . Let n�¬�o SUT V W=XjY ¬\] be such that Rest n � q �Ä t ½ � kRest n � q �Ä t ½ � . Then the map Ú k©ä n � c in

Y �n � c inY � belongs to

SUT V W.

The following results explain the connections between good restrictions and traces.

Lemma B.2 Let _f`�aM` � c � `�eÕ`�bdcuae�¤ � . Let nÇo S{T V W@XZY � ] . Assume that there exists �fkRest n q �Ä t ½ � . Then ��k tr n q Ä t ½ � .

PROOF. Let Ú k ä n ¬ ã�0c inY �_�c inY � t By Lemma B.1, we have Ú o S{T V W=XjY ] . By trace theory and

continuity of the trace, we have _�k tr Ú q Ä t ½ � , so that tr n q Ä t ½ � kg� .

135


Lemma B.3 Let _-`�a�` � c � Òef`gbdc�aef} � . Let n"o S{T � HÂ W7V W|XjY � ] . Then, considered as aS{TLV W

function, n has a good downward restriction to m À i kg_ x which coincides with tr n q Ä t ½ � .

PROOF. Let �,k tr n q Ä t ½ � . Then �¦o SUT V W=X ��] , by the trace theory. By Lemma B.1, it remains to provethat � �� q n XÁÀ ] ¬ ã� XÁÀ ] q WÀ TjWi � À `gb t X�� t® ]Assume first that a ¥d�>â eNk � . Then (B.5) follows from the well-known Hardy inequality�#¯�� H� q n XÁÀ �jc ê ] ¬ n X�À �jc_ö] q Wê W � ê � À ³�~N°±M,n]° W $ Ë cc��nfo S H V W XjY � ] t X�� t ² ]Consider now the case where a ¥d�>â e ük � . Let ó#k:a ¥d�>â e . We are going to prove that� �� q n X�À ] ¬ ã� XÁÀ ] q WÀ TjWi � À ³g~N°ën]° W È�³ Ê Ë X�� tµ´ ]for some convenient equivalent (semi-) norm on

SU÷�V W. It is useful to consider the norm

§ «¶�� i�� ½ H� �� C Ä3�c�R�ZÄ � � � �c� V Ä � � � � �c�AD

q § X�À ¥ z ê v � ] ¬ z § XÁÀ ¥ ê v � ] ¥ § X�À ] q Wê ÷QW � H � À � ê¢¡E£¤ H�Â W X�� t · ](see, e.g., Triebel [24]).For any

À �òo¸� such that n Ä�¹ kgn X�À �Zc , ]Io SÞ÷ V W@X _�c � ] , the map§ Ä ¹ Xjê ]·k¨ä n X�À �jc ê ]Lc ifê ¤�_� X�À � ]Lc ifê `�_

belongs toS�÷�V W@X ¬ � c � ] , by standard trace theory. Moreover, for any such

À � we have°E§ Äb¹ ° W È�³ Ê Ë � �òH V H�� ³Î~N°ën Ä�¹ ° W È�³ Ê Ë � � V H�� c X�� t º ]i.e. � �� C/» � � �òH V H��¼� » � �¢� � �2H V H�� V » � � �¢� � �2H V H/�rD

q § Ä�¹ X¼½ ¥ z ê ] ¬ z § Ä�¹ Xr½ ¥ ê ] ¥ § Ä�¹ Xr½ ] q Wê ÷QW � H � ½ � ê ³~ � �� C�» � � � V H��¼� » � �¢� � � V H�� V » � � �¢� � � V H/�rD � q n Äb¹ Xr½ ¥ z ê ] ¬ z7n Ä�¹ Xr½ ¥ ê ] ¥ n Ä�¹ X¼½ ] q Wê ÷QW � H � ½ � ê t

In particular, � k � HÂ �� u�� q § Äb¹ X¼½ ¥ z ê ] ¬ zD§ Äb¹ X¼½ ¥ ê ] ¥ § Äb¹ X¼½ ] q Wê ÷QW � H � ½ � ê ³g~N°ën Ä�¹ ° W È ÷�V W t X¼� tÓ� _w]Since � }g~ � HÂ¹� q n X�À �Ác ê ] ¬ � X�À ��] q Wê ÷QW � ê kR~ � H�Â ¹� q n XÁÀ �Zc ê ] ¬ � X�À �Á] q Wê TjW � H � ê c X¼� tÓ�D� ]we find that � HÂ ¹� q n X�À �Zc ê ] ¬ � XÁÀ �Á] q Wê TjW � H � ê ³C~N°*n Äb¹ ° W È ³ Ê Ë t X¼� tÓ� zD]On the other hand, we clearly have� HHÂ ¹ q n XÁÀ �jc ê ] ¬ � XÁÀ ��] q Wê TjW � H � ê ³g~N°ën Ä�¹ ° W $ Ë ¥ ~ q � X�À � ] q W t X¼� tÓ� ²w]136


By combining (B.12), (B.13) and integrating with respect toÀ � , we obtain (B.7). The proof of Lemma B.3

is complete.

A simple consequence of Lemma B.3 is the following

Lemma B.4 Let _�`ga�` � c � `"ef`dbdc�a�ef} � and �N¤ga . Let n H o SUT V W=XZY � ] and n � o S¾�^V W=XZY � ] .Assume that n H has a good downward restriction ��k Rest n H q �Ä t ½ � and that �fk tr n � q Ä t ½ � . Then themap Ú k ä n H c in

Y �n0��c inY �

belongs toS T V W XZY ] .

PROOF. Let n ¹ o SUT � HÂ W>V W|XjY � ] be an extension of � . Then Ú k Ú H ¥ Ú � , whereÚ H k¨ä n H c inY �n ¹ c inY �

and Ú � k¨ä _�c inY �n�� ¬ n ¹ c inY � t

By Lemma B.3 and the assumption �#k Rest n H q �Ä t ½ � , we have Rest n H q �Ä t ½ � k Rest n ¹ q �Ä t ½ � . ByCorollary B.3, we find that Ú H o SUT V W@XZY ] . It remains to prove that Ú ��o SUT V W=XjY ] . Let ó¸k minm��c a ¥d�>â e3c � x . Then Ú ��o SÞ÷�V W=XjY ] , by standard trace theory. Thus Ú ��o SUT V W=XjY ] .

We conclude this section by stating the following precised form of Corollary B.1, b) in the case of ageneral boundary. We use the same notations as in the proof of Theorem 1, Case 4.

Lemma B.5 em Let nfo S H�Â W7V W@XZY ] . Thena) for a.e. _-`�ÖB`�Ù we haven q � � o S H�Â W7V W X � � ] and

� � � � � q n X�À ] ¬ n X� ] q Wq À ¬ 2q i � H�� a Ä `gb [ X¼� tÓ��¨ ]b) for any such Ö , n has a good restriction to �� which coincides (a.e. on �� ) with n q � �

.

Appendix C Global lifting

In this appendix, we investigate the existence of a global lifting in some domains with non-trival topology.

Lemma C.1 Let _�`Þa�`Þbgc � `�eO`ÞbdcuaeÔ}R°#c°õ}Þz . Let nÔo SUT V W=X G\H� #¿ H [ G\H^] be such thatdeg

X n q 4 � :&I � ]lkR_ . Then there is some � o S T V W X G H �¿ H [ G H ] such that nMk v �� .

Here, ¿ H is the unit ball in hliª�2H .PROOF. Let � � h! �¿ H GIH>c�� XÁê c À ]\kgn X vQ� �Lc À ] . Then �No S T V Wæyçè X h! �¿ H [ GIHE] , where “loc” refers onlyto the variable

ê. By Theorem 2 in Bourgain - Brezis - Mironescu [4], there is some Sgo S T V Wæyç è X hP N¿ H [ h·]

such that �Kk v>�UT . We claim that S is zJ2 -periodic in the variableê. Indeed, for a.e.

À od¿ H , we haven�o SUT V W|X G\HL �m À x [ GIH*] and degX n q 4 � :�C ÄJD ]ªk�_ . In particular, for any such

Àthe map n q 4 � :&C ÄJD has a

continuous lifting ¿ Ä . On the other hand, for a.e.À oC¿ H we have S Ä kÀS X-, c À ]Bo S TLV Wæyç è X h� Km À x [ h·] .

Thus, with Ä XÁê ]�kÁ¿ Ä X v>� �] , we find that for a.e.À od¿ H the function S Ä ¬ Ä is continuous and zJ2RX

-valued; therefore it is a constant. Since Ä is z32 -periodic, so is S Ä for a.e.À o�¿ H . We obtain that S iszJ2 -periodic in the variable

ê. Thus the map � � G�H9 O¿ H h�c � X vQ� �ëc À ]�kÂS XÁê c À ] is well-defined and

belongs toS{T V W@X G\H� �¿ H [ h·] . Moreover, we clearly have nMk v��¶� .

In the same vein, we have

137


Lemma C.2 Let a,} � c � `Õe�Ùbgc�°(}:²�cuzM³Þae�`:° . Let nKo S{T V W@X G\H� #¿ H [ G\HE] be such thatdeg

X n q 4 � :ÃI � ]lkd_ . Then there is some � o S{TLV W@X GIHÄ �¿ H [ h·] ¯�S H V TjW�X G\Hd -¿ H [ h·] such that nMk v>�� .

The proof is similar to that of Lemma C.1; one has to use Lemma 4 in [4] instead of Theorem 2 in [4].

Lemma C.3 Let � `"ef`db and Ö H ¤C_ . Then there is some Ö � ¤C_ such that every �No S HÂ W>V W|X GIH [ GIHE]satisfying °*� ¬ � ° È �¼Å Ë*Ê Ë �

4� � `�Ö � has a global lifting � o S HÂ W>V W=X G\H [ h·] such that ° � ° È �rÅ Ë*Ê Ë �

4� � `CÖ H .

PROOF. Recall that if

�is an interval, then every Ú o S H�Â W7V W@X � [ G\H^] has a lifting S4o S H�Â W7V W@X � [ h\]

(see Bourgain - Brezis - Mironescu [4], Theorem 1). Moreover, this lifting may be chosen to be (locally)continuous with respect to Ú , i.e. for every Ú � o S HÂ W>V W�X � [ G\HE] there is some Ö � ¤Ç_ such that in the setm Ú [ ° ÚC¬OÚ � ° È �rÅ Ë*Ê Ë �UÆ �

4� � `�Ö � x

there is a lifting ÚR« S continuous for theS H�Â W7V W norm. (This assertion can be established using the same

argument as in Step 7 of the proof of Theorem 4 in Brezis, Nirenberg [12]; it can also be derived from theexplicit construction of S in the proof of Theorem 1 in [4]; see also Boutet de Monvel, Berthier, Georgescu,Purice [6] when e�k:z ).Let

� k ´ ¬ z32lc zJ20µ . To each �No S H�Â W7V W@X G\H [ G\HE] we associate the map Ú o S HÂ W7V W@X � [ GIH^] , Ú Xjê ]�kà� X v>� �] .By the above considerations, for every Ö ¹ ¤�_ there is some Ö±Ç�¤g_ such that, if °*� ¬ � ° È �rÅ Ë*Ê Ë �

4� � `õÖ±Ç ,

then Ú has a lifting S such that °ES�° È �¼Å Ë*Ê Ë ��Æ � `:Ö ¹ . We claim that S is z32 -periodic if Ö ¹ is small enough.Indeed, the function È XÁê ]JkÉS Xjê ¬ z32s] ¬ S Xjê ] belongs to

S HÂ W7V W�X ´ _�cLzJ20µ [ zJ2RXJ] , so that È is constant a.e.(see [4], Theorem B.1). Since °EÈ&° $ � ³�°ÊS6° $ � `g~�Ö ¹ , we have È�kR_ (i.e. S is zJ2 -periodic) if ~�Ö ¹ `gzJ2 .

Thus, for Ö ¹ small enough, the map � X v � � ]�kjS Xjê ] is well-defined, belongs toS HÂ W>V W and satisfies° � ° È �¼Å ËEÊ Ë �

4� � `�Ö H and n¦k v>�� .

Appendix D Filling a hole - the fractional case

We adapt to fractional Sobolev spaces the technique of Brezis - Li [7], Section 1.3.The first two results are preparations for the proofs of Lemmas 5,6 and 8 (see Lemmas D.3, D.4 and

D.5 below).

Lemma D.1 Let _Õ`þaO` � c � `{e�`ôbgc � `þaeÞ`4° . Let ~±k X ¬ � c � ]�i and nÞo SUT V W|X î2~�] .Then

ãnKo SUT V W=X ~�] ; here,ãn X�À ]=k�n X�À â q ÀFq ] and

q�qis the

× � norm in h·i . Moreover, the map n « ãn iscontinuous from

S{T V W@X î0~�] intoSUT V W=X ~�] .

PROOF. Clearly, we have ° ãn`° $ Ë �fQ � ³£~ � °*nË° $ Ë � � Q � . Thus it suffices to prove, for the Gagliardo semi-norms in

SUTLV W, the inequality ° ãn`° W È É8Ê Ë �fQ � ³�~ H X °ën]° W È É/Ê Ë � � Q � ¥ °ën]° W $ Ë � � Q � ] t X�Ì tÓ� ]

We have�Q�Q

q ãn XÁÀ ] ¬ ãn X ] q Wq À ¬ òq i � TZW � À � k � H� � H� �� Q�� Q

q n XÁÀ ] ¬ n X� ] q Wq 7 À ¬ ó òq i � TjW 7 i|�òH ó iª�òH � a Ä � abW � 7 � ó t X�Ì t zw]We claim that � k � H� � H� 7 iª�2H ó iª�òHq 7 À ¬ ó òq i � TZW � 7 � ó�³C~�� â q À ¬ òq i � TZW t X�Ì t ²ö]Indeed, � k � H� � HÂ/Í� 7�iª�2H X &7u]/iª�2Hq 7 À ¬ Ã7 òq i � TjW � � 7 k� H� � HÂ/Í� 7 iª� TjW �2H �i|�òHq À ¬ òq i � TZW � � 7f³ � H ¥ � � c X�Ì t ¨ ]138


where

� H k�O H� O �� and

� � k~O H� O �� .On the one hand, we have� H k � H� � �� 7 iª� TZW �òH uiª�2Hq À ¬ òq i � TjW � � 7³�~ ¹ � H� � �� 7 iª� TjW �2H iª�2Hq À ¬ òq i � TjW � � 7�³�~dÇ â q À ¬ òq i � TjW t X�Ì t ]

On the other hand, we have� � k � H� � �� 7 iª� TjW �òH �iª�2Hq À ¬ òq i � TjW � � 7³g~ÄÎ � H� � �� 7 iª� TZW �òH �iª�2H i � TjW � � 7Mk:~ÄÎ � H� � �� 7 iª� TZW �òH � TZW �òH � � 7f³g~dÏ t X�Ì t ² ]We obtain (D.3) by combining (D.4), (D.5) and (D.6). Finally, (D.1) follows from (D.2) and (D.3).

The proof of Lemma D.1 is complete.

Lemma D.2 Let _R`±a�` � c � `4eÍ`±bdc � `Ìae£`±° . Let ��c Ú o S T V W X ~ [ G H ] be such that� q � Q k Ú q � Q o SUT V W@X î0~�] . Then, there is a homotopy éÍo�~�� X ´ _�c � µ [LSUT V W@X ~ [ G\H^]�] such that é X _�c , ]\k�ucué X � c , ]lk Ú and é XÁê c , ] q � Q kg� q � Q cÐ� ê o ´ _�c � µ .PROOF. Let nfk:� q � Q . It clearly suffices to prove the lemma in the special case Ú k ãn . In this case, let,for _-³ ê ` � , é XÁê c À ]·k ä � X�À â X � ¬ ê ]]Lc if

q À�q ³ � ¬ êãn X�À ]Lc if � ¬ ê ` q À�q ³ � [set é X � c , ]�k ãn . Clearly, é{o#~J� X ´ _�c � ] [LSUT V W@X ~ [ G\H^]�] . It remains to prove that é Xjê c , ] ãn as

ê � . Let§ X�À ]lk ä � X�À ]Lc ifq À�q ³ �ãn X�À ]ëc ifq À�q ¤ �

and Ñ�kR§ ¬ ãn . Then §�c ãn"o S T V WæÓçè X hli,] , so that Ñ�o S T V WæÓçè X h·i,] . Since Ñ-kg_ outside ~ , we actually haveÑNo SUT V W@X h�i,] . Thus °^é Xjê c , ] ¬ ãn]° W È É8Ê Ë �fQ � k�°(Ñ X., â X � ¬ ê ]�]±° W È É/Ê Ë �fQ � ³°(Ñ X., â X � ¬ ê ]�]±° W È É/Ê Ë �µÒ t � k X � ¬ ê ] iª� TjW °(ÑR° W È É/Ê Ë �µÒ t � _as

ê � . The proof of Lemma D.2 is complete.

We introduce a useful notation: let n#o S�T � V W � X n * ] , where _-`�a H ` � c � `Ôe H `gbdc � `ga H e H `�° .We extend, for each ~ o n * � H cn q � Q to ~ as in Lemma D.1. Let

ãn be the map obtained by gluing theseextensions. We next extend

ãn ton * � � in the same manner, and so on, until we obtain a map defined in

n i ;call it ý * X n2] .Lemma D.3 Let _R`¾a H ` � c � `ë H `Ìbgc � `àa H e H `Î°#c ´ a H e H µ�³Óq:`¾° . Then every ��oSUT � V W � X n � [ G\H*] has an extension n H o SUT � V W � X n i [ G\HE] such that n H q s�� o SUT � V W � for pòk�q c tÓt¶t c° ¬ � .

PROOF. We take n H k£ý�� X ��] . We may use repeatedly Lemma D.1, since for pIk¾q ¥Þ� c tÓt¶t c° we have� `�a H e H `�p .Lemma D.4 Let _Õ`Îa�` � c � `£e:` bgc � `Îa�e�`4°#c ´ ae�µ�³�q�`þ° . If n q s � o SUT V W c�n H q s � oSUT V W c�pòkÔqDc tÓt¶t c ° ¬ � , and n q s � k�n H q s � , then n and n H are

SUT V W-homotopic.

PROOF. We argue by backward induction on q . If qRk ° ¬ � , then for each ~ o n i Lemma D.2provides a

S T V W-homotopy of n q Q and n H q Q preserving the boundary condition. By gluing together these

139


homotopies we find that n and n H areS{T V W

-homotopic (here we use �>â eR`4a�` � ). Suppose now thatthe conclusion of the lemma holds for q ¥U� ; we prove it for q , assuming that qK} ´ a�e�µ . By assumption,n and ý�� H X n q s �

�� ] are

SUT V W-homotopic, and so are n H and ý�� H X n H q s �

�� ] . It suffices therefore to prove

that �:k±ý�� H X n q s �� ] and � H kºý�� H X n H q s �

�� ] are

S{T V W-homotopic. For each ~Ïo n � � H , we have� q � Q kU� H q � Q k�n q � Q kUn H q � Q . By Lemma D.2, � q Q and � H q Q are connected by a homotopy preserving

the trace on î2~ . Gluing together these homotopies, we find that � q s �� and � H q s �

�� are

SUT V W-homotopic.

If é connects � q s �� to � H q s �

�� , then Lemma D.1 used repeatedly implies that

ê « ý�� H X é Xjê ]�] connectsin

SUT V W@X n i [ G\HE] the map ý�� H X � q s �� ] to ý�� H X � H q s �

�� ] , i.e., � to � H .

The proof of Lemma D.4 is complete.

Lemma D.5 Let _�`{a,` � c � `CeK`�bgc � ÙaeK`Þ°#c ´ ae�µI³Pq#`�° . Let �uc Ú o SUT V W|X n i [ G\H*] besuch that � q s�� o SUT V W c Ú q s�� o SUT V W c�p|k�qDc tÓt¶t c ° ¬ � . Assume that � q s � and Ú q s � are

SUT V W-homotopic.

Then � and Ú areS T V W

-homotopic.PROOF. By Lemma D.4, � and ýB� X � q s � ] (respectively Ú and ý � X Ú q s � ] ) are

SUT V W-homotopic. If é con-

nects � q s � to Ú q s � inSUT V W

, then as in the proof of Lemma D.4, we obtain thatê « ý�� X é Xjê ]�] connectsý � X � q s � ] to ý � X Ú q s � ] in

SUT V W. Thus � and Ú are

S{T V W-homotopic.

Appendix E Slicing with norm control

In this section, we prove the existence of good coverings forS£T V W

maps. The arguments are rather standard.Without loss of generality, we may consider maps defined in h\i . Throughout this section, we assumeÙ�k � , i.e. we consider a covering with cubes of size 1. We start by introducing some useful notations: forÀ o�~ i k X _�c � ] i and for q�k � c tÓt¶t c° ¬ � , let

~·�Jk�o ä ��* ½ H ê * v � } ¥ iª� �� æ ½ H æ v � � [�ê * o�h�c( æ o"X�c*m v � } x ð m v � � x k�m v H c t¶tÓt v i x�Õand ~ � X�À ]·k À ¥ ~ � . (With the notations introduced in Section 3, we have ~ � X�À ]·k n Ä� when

Y kRhli ).For a fixed set

ñ Ý:m � c t¶t c ° x such thatq ñ�q kÔq , let also

~ ;� k ä � � � ; ê � v � ¥ �� Â� ; ö� v � [�ê � o�h�c/ ��No!X Õ cso that ~·�Jk ð m>~ ;� [ ñ ÝRm � c tÓt¶t c ° x c q ñ�q kÔq x cand with obvious notations ~ � XÁÀ ]Ik ð m>~ ;� XÁÀ ] [ñ Ý:m � c t¶tÓt c° x c q ñJq k�q xDt

Instead of considering a fixed (semi-) norm onS£T V W c _�`¾aK` � c � `Íe{`¾b , it is convenient to

consider a family of equivalent normsq § q W� k �ñ ÝRm � c t¶t¶t c ° xq ñ�q k§q� Ò t � Ò � q § XÁÀ ¥�Ö � � ; ê � v � ] ¬ § X�À ] q Wq ê*q � � TjW � ê � À

(see, e.g., Triebel [24]). An obvious computation yields, for the usual Gagliardo (semi-) norm on ~ ;� X�À ] ,140


Lemma E.1 Let _�`ga�` � c � `Ôef`db and nfo S{T V W. Then�ñ ÝRm � c t¶t¶t c ° xq ñ�q kÔq

� Q t °*nË° W È É/Ê Ë �UQA×� � Ä^��¢� À ³ q n q W� X¼Ø t¶� ]for some ~ independent of n .

We next define the norm °*nË° È É/Ê Ë �UQ � � ÄQ�� by the formula°*nË° W È É/Ê Ë �UQ � � Ä^�� k �Q � Q �� ÄQ� °ën]° W È É/Ê Ë � � Q � t

Lemma E.2 Let _�`ga�` � c � `Ôef`db . Then, for n#o S{T V W, we have

a) for a.e.À o�~�iBc�n q Q � � ÄQ� o S T V WæÓçè crqBk � c tÓt¶t c° ¬ � ;

b) there is a fat set (i.e., with positive measure) ÙUÝC~Bi such that°*nË° W È É8Ê Ë �fQ � � ÄQ�� ³�~ q n q W� c�� À o"Ù t X¼Ø t zD]Remark E.1. Here, n q Q � � Ä^� are restrictions, not traces. However, when aef¤ � we may replace restrictionsby traces, by a standard argument. We obtain

Corollary E.1 Let _R` aÕ` � c � `Íe�`¾bdcuae{¤ � . Let n�o S T V W. Then, for a.e.

À o�~ i , trn q Q tvu � � Ä^� o SUTLV W. Moreover, for a.e.

À oÕ~�i , tr n q Q tvu � � ÄQ� has a trace on ~ iª� � XÁÀ ] which belongs toSUT V W, and so on.

PROOF OF LEMMA E.2. In order to avoid long computations, we treat only the case q¦k � c °õk£z . Thegeneral case does not bring any additional difficulty. Let ~ o�~ H X�À ] ; denote its lower (resp. upper, left,right) edge by ~ æ

(resp. ~6Ú�cu~ $ c�~�Û ). By (E.1), we have n q Q � o SUT V Wfor a.e.

À o"~ � and, forÀ

in a fatset, Ö Q � Q2�� ÄQ� °ën]° W È É/Ê Ë �fQ � � ³ const.

q n q W H . Similar statements hold for the other edges.It remains to control the cross - integrals in the Gagliardo norm, e.g. to prove� k � Q � �Q � Q0� � Ä^� � Q � � QvÜ

q n X� ] ¬ n X�Ý ] q Wq ¬ Ý�q � � TjW � � Ý ³ const. °*nË° W È É/Ê Ë X¼Ø t ²w](here, we take the usual Gagliardo norm in

S�T V W@X h � ] ). We have� k � Q � �Þ �3ß � � H� � H� q n XÁÀ ¥§à H v H ¥§à � v � ¥ 7 v H ] ¬ n XÁÀ ¥áà H v H ¥§à � v � ¥ ó v �>] q Wq 7 v H ¬ ó v � q � � TjW � ó � 7 � Àk � Ò � � H� � H� q n X ¥ 7 v H ] ¬ n X ¥ ó v � ] q Wq 7 v H ¬ ó v � q � � TjW � ó � 7 � k � Ò � � H� � H� q n X�Ý ] ¬ n X�Ý ¬ 7 v H ¥ ó v � ] q Wq 7 v H ¬ ó v � q � � TjW � ó � 7 � Ý³ � Ò � � Ò � q n X¼Ý ¥ ½ ] ¬ n X¼Ý ] q Wq ½Fq � � TjW � ½ � Ý k�°*nË° W È É8Ê Ë tThe proof of Lemma E.2 is complete.

Acknowledgement. The first author (H.B.) warmly thanks Yanyan Li for useful discussions. He ispartially supported by a European Grant ERB FMRX CT980201, and is also a member of the InstitutUniversitaire de France. This work was initiated when the second author (P.M.) was visiting RutgersUniversity; he thanks the Mathematics Department for its invitation and hospitality. It was completedwhile both authors were visiting the Isaac Newton Institute in Cambridge, which they also wish to thank

141


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H. Brezis P. MironescuANALYSE NUMERIQUE DEPARTEMENT DE MATHEMATIQUESUNIVERSITE P. ET M. CURIE, B.C. 187 UNIVERSITE PARIS-SUD4, Pl. Jussieu 91405 ORSAY 75252PARIS CEDEX 05 [email protected]@ccr.jussieu.frRUTGERS UNIVERSITYDEPT. OF MATH., HILL CENTER, BUSCH CAMPUS110 FRELINGHUYSEN RD, PISCATAWAY, NJ [email protected]

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On some questions of topology for -valued fractional Sobolev spaces

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